John Stafford

Professor John Stafford

Research Fellow

Organisation

University of Manchester

Research summary

Nature is inherently noncommutative: applying operations (like rotations and reflections or even putting on one's shoes and socks) in different orders leads to different outcomes. A classic example from quantum physics is Heisenberg's Uncertainty Principle which after rescaling is codified by the equation qp - pq = 1. Noncommutative algebra seeks to understand these (and more sophisticated) algebraic concepts. For example, the Weyl Algebra or Algebra of Quantum Mechanics is the set of all ``polynomials" in the variables p and q, where we use the rule qp = 1 + pq to multiply polynomials.

My research interests are primarily in noncommutative algebra, most especially in noncommutative algebraic geometry: an area that has flourished recently with increasing applications to other areas of mathematics and physics. Algebraic geometry is one of the deepest areas of mathematics; it analyzes solutions of polynomial equations and their associated geometric constructs. For example, solutions of x^2 - y^3 = 0 in the plane define a curve called a cusp. Noncommutative algebraic geometry is concerned with the interaction between (projective) algebraic geometry and noncommutative algebra and is very successful in using algebraic geometry to understand noncommutative structures. A major problem that I and my students are working on is to classify and understand noncommutative projective surfaces and we have made recent substantial progress.

Grants awarded

Noncommutative Algebraic Geometry and its Applications

Noncommutative Algebraic Geometry and its Applications

Scheme: Wolfson Research Merit Awards

Dates: Oct 2007 - Mar 2013

Value: £100,000

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